Critical exponents of planar gradient percolation
Pierre Nolin
Source: Ann. Probab. Volume 36, Number 5 (2008), 1748-1776.
Abstract
We study gradient percolation for site percolation on the triangular lattice. This is a percolation model where the percolation probability depends linearly on the location of the site. We prove the results predicted by physicists for this model. More precisely, we describe the fluctuations of the interfaces around their (straight) scaling limits, and the expected and typical lengths of these interfaces. These results build on the recent results for critical percolation on this lattice by Smirnov, Lawler, Schramm and Werner, and on the scaling ideas developed by Kesten.
Primary Subjects: 60K35, 82B43, 82B27
Keywords: Inhomogeneous percolation; gradient percolation; critical exponents; random interface
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
This document is available for purchase at a cost of $15. Select the "buy article" button below to make a credit card purchase of this document through a secure payment site.
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1221138765
Digital Object Identifier: doi:10.1214/07-AOP375
Mathematical Reviews number (MathSciNet):
MR2440922
References
Aizenman, M., Duplantier, B. and Aharony, A. (1999). Path crossing exponents and the external perimeter in 2D percolation. Phys. Rev. Lett. 83 1359--1362.
Beffara, V. and Sidoravicius, V. (2006). Percolation theory. In Encyclopedia of Mathematical Physics. Elsevier, Amsterdam. To appear.
Mathematical Reviews (MathSciNet):
MR2198848
Zentralblatt MATH:
1064.60194
Borgs, C., Chayes, J. T., Kesten, H. and Spencer, J. (1999). Uniform boundedness of critical crossing probabilities implies hyperscaling. Random Structures Algorithms 15 368--413.
Mathematical Reviews (MathSciNet):
MR1716769
Borgs, C., Chayes, J. T., Kesten, H. and Spencer, J. (2001). The birth of the infinite cluster: Finite-size scaling in percolation. Comm. Math. Phys. 224 153--204.
Mathematical Reviews (MathSciNet):
MR1868996
Digital Object Identifier: doi:10.1007/s002200100521
Camia, F. and Newman, C. M. (2006). Two-dimensional critical percolation: The full scaling limit. Comm. Math. Phys. 268 1--38.
Mathematical Reviews (MathSciNet):
MR2249794
Digital Object Identifier: doi:10.1007/s00220-006-0086-1
Camia, F., Fontes, L. R. G. and Newman, C. M. (2006). The scaling limit geometry of near-critical 2D percolation. J. Stat. Phys. 125 1159--1175.
Mathematical Reviews (MathSciNet):
MR2282484
Digital Object Identifier: doi:10.1007/s10955-005-9014-6
Camia, F., Fontes, L. R. G. and Newman, C. M. (2006). Two-dimensional scaling limits via marked nonsimple loops. Bull. Braz. Math. Soc. 37 537--559.
Mathematical Reviews (MathSciNet):
MR2284886
Digital Object Identifier: doi:10.1007/s00574-006-0026-x
Chayes, J. T., Chayes, L. and Fröhlich, J. (1985). The low-temperature behavior of disordered magnets. Comm. Math. Phys. 100 399--437.
Mathematical Reviews (MathSciNet):
MR0802552
Digital Object Identifier: doi:10.1007/BF01206137
Project Euclid: euclid.cmp/1104113922
Chayes, J. T., Chayes, L., Fisher, D. S. and Spencer, T. (1986). Finite-size scaling and correlation lengths for disordered systems. Phys. Rev. Lett. 57 2999--3002.
Mathematical Reviews (MathSciNet):
MR0925751
Digital Object Identifier: doi:10.1103/PhysRevLett.57.2999
Desolneux, A., Sapoval, B. and Baldassarri, A. (2004). Self-organized percolation power laws with and without fractal geometry in the etching of random solids. In Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 2 (M. L. Lapidus and M. van Frankenhuijsen, eds.) 485--505. Amer. Math. Soc. Providence, RI.
Mathematical Reviews (MathSciNet):
MR2112129
Zentralblatt MATH:
1067.60097
Grimmett, G. R. (1999). Percolation, 2nd ed. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1707339
Kesten, H. (1980). The critical probability of bond percolation on the square lattice equals 1$/$2. Comm. Math. Phys. 74 41--59.
Mathematical Reviews (MathSciNet):
MR0575895
Digital Object Identifier: doi:10.1007/BF01197577
Project Euclid: euclid.cmp/1103907931
Kesten, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet):
MR0692943
Zentralblatt MATH:
0522.60097
Kesten, H. (1987). Scaling relations for 2D-percolation. Comm. Math. Phys. 109 109--156.
Mathematical Reviews (MathSciNet):
MR0879034
Digital Object Identifier: doi:10.1007/BF01205674
Project Euclid: euclid.cmp/1104116714
Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 237--273.
Mathematical Reviews (MathSciNet):
MR1879850
Digital Object Identifier: doi:10.1007/BF02392618
Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 275--308.
Mathematical Reviews (MathSciNet):
MR1879851
Digital Object Identifier: doi:10.1007/BF02392619
Lawler, G. F., Schramm, O. and Werner, W. (2002). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7 1--13.
Mathematical Reviews (MathSciNet):
MR1887622
Nolin, P. (2007). Near-critical percolation in two dimensions. Preprint. arXiv:0711.4948.
Rosso, M., Gouyet, J. F. and Sapoval, B. (1985). Determination of percolation probability from the use of a concentration gradient. Phys. Rev. B 32 6053--6054.
Sapoval, B., Rosso, M. and Gouyet, J. F. (1985). The fractal nature of a diffusion front and the relation to percolation. J. Phys. Lett. 46 L149--L156.
Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221--288.
Mathematical Reviews (MathSciNet):
MR1776084
Digital Object Identifier: doi:10.1007/BF02803524
Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239--244.
Mathematical Reviews (MathSciNet):
MR1851632
Digital Object Identifier: doi:10.1016/S0764-4442(01)01991-7
Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729--744.
Mathematical Reviews (MathSciNet):
MR1879816
Van den Berg, J. and Kesten, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556--569.
Mathematical Reviews (MathSciNet):
MR0799280
Digital Object Identifier: doi:10.2307/3213860
JSTOR: links.jstor.org
Ziff, R. M. and Sapoval, B. (1986). The efficient determination of the percolation threshold by a frontier-generating walk in a gradient. J. Phys. A: Math. Gen. 19 L1169--L1172.
The Annals of Probability
